Three circles, $a$, $b$, and $c$ have radii $r_a = f(r_c)$, $r_b = 1$, and $r_c, (r_c < r_a \le r_b)$, respectively, and are externally tangent to each other and a line. Find: $$\sum\limits_{k=2}^{100} \frac{1}{f\left(\frac{1}{k^2}\right)}$$